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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|VALUING CASH FLOWS

Consider times $t_1$ and $t_2$, where $t_2$ is not necessarily greater than $t_1$. The value at time $t_1$ of the sum $C$ due at time $t_2$ is defined as follows.
(a) if $t_1 \geq t_2$, the accumulation of $C$ from time $t_2$ until time $t_1$, or
(b) if $t_1t_2, \int_{t_1}^{t_2} \delta(t) \mathrm{d} t=-\int_{t_2}^{t_1} \delta(t) \mathrm{d} t$.
Since
$$\int_{t_1}^{t_2} \delta(t) \mathrm{d} t=\int_0^{t_2} \delta(t) \mathrm{d} t-\int_0^{t_1} \delta(t) \mathrm{d} t$$
it follows immediately from Eqs $2.5 .3$ and $2.7 .1$ that the value at time $t_1$ of $C$ due at time $t_2$ is
$$C \frac{v\left(t_2\right)}{v\left(t_1\right)}$$
The value at a general time $t_1$, of a discrete cash flow of $c_t$ at time $t$ (for various values of $t$ ) and a continuous payment stream at rate $\rho(t)$ per time unit, may now be found, by the methods given in Section 2.6, as
$$\sum c_t \frac{v(t)}{v\left(t_1\right)}+\int_{-\infty}^{\infty} \rho(t) \frac{v(t)}{v\left(t_1\right)} \mathrm{d} t$$
where the summation is over those values of $t$ for which $c_t \neq 0$. We note that in the special case when $t_1=0$ (the present time), the value of the cash flow is
$$\sum c_t v(t)+\int_{-\infty}^{\infty} \rho(t) v(t) \mathrm{d} t$$
where the summation is over those values of $t$ for which $c_t \neq 0$. This is a generalization of Eq. 2.6.7 to cover past, as well as present or future, payments.
If there are incoming and outgoing payments, the corresponding net value may be defined, as in Section 2.6, as the difference between the value of the positive and the negative cash flows. If all the payments are due at or after time $t_1$, their value at time $t_1$ may also be called their discounted value, and if they are due at or before time $t_1$, their value may be referred to as their accumulation. It follows that any value may be expressed as the sum of a discounted value and an accumulation; this fact is helpful in certain problems. Also, if $t_1=0$, and all the payments are due at or after the present time, their value may also be described as their (discounted) present value, as defined by Eq. 2.6.7.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|INTEREST INCOME

Consider now an investor who wishes not to accumulate money but to receive an income while keeping his capital fixed at $C$. If the rate of interest is fixed at $i$ per time unit, and if the investor wishes to receive his income at the end of each time unit, it is clear that his income will be iC per time unit, payable in arrears, until such time as he withdraws his capital.

More generally, suppose that $t>t_0$ and that an investor wishes to deposit $C$ at time $t_0$ for withdrawal at time $t$. Suppose further that $n>1$ and that the investor wishes to receive interest on his deposit at the $n$ equally spaced times $t_0+h, t_0+2 h, \ldots, t_0+n h$, where $h=\left(t-t_0\right) / n$. The interest payable at time $t_0+(j+1) h$, for the period $t_0+j h$ to $t_0+(j+1) h$, will be
$${ }^C{ }^2 i_h\left(t_0+j h\right)$$
where $i_h(t)$ is the nominal rate over the period $h$ starting at time $t$. The total interest income payable between times $t_0$ and $t$ will then be
$$C \sum_{j=0}^{n-1} h i_h\left(t_0+j h\right)$$
Since, by assumption, $i_h(t)$ tends to $\delta(t)$ as $h$ tends to 0 , it is fairly easily shown (provided that $\delta(t)$ is continuous) that as $n$ increases (so that $h$ tends to 0 ) the total interest received between times $t_0$ and $t$ converges to
$$I(t)=C \int_{\mathrm{t}_0}^t \delta(s) \mathrm{ds}$$
Hence, in the limit, the rate of payment of interest income per unit time at time $t$, $I^{\prime}(t)$, equals
$C \delta(t)$
The position is illustrated in Figure 2.8.1. The cash $C$ in the “tank” remains constant at $C$, while interest income is decanted continuously at the instantaneous rate $C \delta(t)$ per unit time at time $t$. If interest is paid very frequently from a variable-interest deposit account, the position may be idealized to that shown in the figure, which depicts a continuous flow of interest income. Of course, if $\delta(t)=\delta$ for all $t$, interest is received at the constant rate Cô per time unit.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|VALUING CASH FLOWS

(a) 如果 $t_1 \geq t_2$, 的积㽧 $C$ 从时间 $t_2$ 直到时间 $t_1$, 或
(b) 如果 $t_1 t_2, \int_{t_1}^{t_2} \delta(t) \mathrm{d} t=-\int_{t_2}^{t_1} \delta(t) \mathrm{d} t$.

$$\int_{t_1}^{t_2} \delta(t) \mathrm{d} t=\int_0^{t_2} \delta(t) \mathrm{d} t-\int_0^{t_1} \delta(t) \mathrm{d} t$$

$$C \frac{v\left(t_2\right)}{v\left(t_1\right)}$$

$$\sum c_t \frac{v(t)}{v\left(t_1\right)}+\int_{-\infty}^{\infty} \rho(t) \frac{v(t)}{v\left(t_1\right)} \mathrm{d} t$$

$$\sum c_t v(t)+\int_{-\infty}^{\infty} \rho(t) v(t) \mathrm{d} t$$

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|INTEREST INCOME

$${ }^{C 2} i_h\left(t_0+j h\right)$$

$$C \sum_{j=0}^{n-1} h i_h\left(t_0+j h\right)$$

$$I(t)=C \int_{\mathrm{t}_0}^t \delta(s) \mathrm{ds}$$

$C \delta(t)$

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